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Introduction to Polygons

- Different types of Polygons
- Simple Convex
- Simple Concave
- Non-simple self-intersecting
- With holes

Convex

Concave

Self-intersecting

Introduction to Polygons

- Convex
- A region S is convex iff for any x1 and x2

in S, the straight line segment connecting x1 and

x2 is also contained in S. The convex hull of an

object S is the smallest H such that S

Scan Line Polygon Fill Algorithms

- A standard output primitive in general graphics

package is a solid color or patterned polygon

area - There are two basic approaches to filling on

raster systems. - Determine overlap Intervals for scan lines that

cross that area. - Start from a given interior point and paint

outward from this point until we encounter the

boundary - The first approach is mostly used in general

graphics packages, however second approach is

used in applications having complex boundaries

and interactive painting systems

Xk1,yk1

Scan Line yk 1

Scan Line yk

Xk , yk

Seed Fill Algorithm

- These algorithms assume that at least one pixel

interior to a polygon or region is known - Regions maybe interior or boundary defined

Interior-defined region

Interior-defined region

A Simple Seed Fill Algorithm

- Push the seed pixel onto the stack
- While the stack is not empty
- Pop a pixel from the stack
- Set the pixel to the required value
- For each of the 4 connected pixels
- Adjacent to the current pixel, check if it

is a boundary pixel or if it has already been set

to the required value. - In either case ignore it. Otherwise push it

onto the stack - The algorithm can be implemented using 8

connected pixels - It also works with holes in the polygons

Scan Line Polygon Fill Algorithm

10 14 18 24

Interior pixels along a scan line passing through

a polygon area

- For each scan line crossing a polygon are then

sorted from left to right, and

the corresponding frame buffer positions between

each intersection pair are set to the specified

color. - These intersection points are then sorted from

left to right , and the corresponding frame

buffer positions between each intersection pair

are set to specified color

Scan Line Polygon Fill Algorithm

- In the given example ( previous slide) , four

pixel intersections define stretches from x10 to

x14 and x18 to x24 - Some scan-Line intersections at polygon vertices

require special handling - A scan Line passing through a vertex intersects

two polygon edges at that position, adding two

points to the list of intersections for the scan

Line - In the given example , scan Line y intersects

five polygon edges and the scan Line y

intersects 4 edges although it also passes

through a vertex - y correctly identifies internal pixel spans ,but

need some extra processing

Scan line Polygon Fill Algorithm

- One way to resolve this is also to shorten some

polygon edges to split those vertices that should

be counted as one intersection - When the end point y coordinates of the two edges

are increasing , the y value of the upper

endpoint for the current edge is decreased by 1 - When the endpoint y values are monotonically

decreasing, we decrease the y coordinate of the

upper endpoint of the edge following the current

edge

Scan Line Polygon Fill Algorithm

(a)

(b)

Adjusting endpoint values for a polygon, as we

process edges in order around the polygon

perimeter. The edge currently being processed is

indicated as a solid like. In (a), the y

coordinate of the upper endpoint of the current

edge id decreased by 1. In (b), the y coordinate

of the upper end point of the next edge is

decreased by 1

Scan Line Polygon Fill Algorithm

- The topological difference between scan Line y

and scan Line y'

- The topological difference between scan line y

and scan line y is - For Scan line y, the two intersecting edges

sharing a vertex are on opposite sides of the

scan line ! - But for scan line y, the two intersecting

edges are both above the scan line - Thu, the vertices that require additional

processing are those that have connecting edges

on opposite sides of scan line. - We can identify these vertices by tracing around

the polygon boundary either in clock-wise or

anti-clockwise order and observing the relative

changes in vertex y coordinates as we move from

one edge to the next. - If the endpoint y values of two consecutive

edges monotonically increase or decrease, we need

to count the middle vertex as a single

intersection point for any scan line passing

through that vertex.

- Otherwise, the shared vertex represents a local

extremum (min. or max.) on the polygon boundary,

and the two edge intersections with the scan line

passing through that vertex can be added to the

intersection list

Figure 3-36 Intersection points along the scan

lines that intersect polygon vertices. Scan line

y generates an odd number of intersections, but

scan line y generates an even number of

intersections that can be paired to identify

correctly the interior pixel spans.

- The scan conversion algorithm works as follows
- Intersect each scanline with all edges
- Sort intersections in x
- Calculate parity of intersections to determine

in/out - Fill the in pixels
- Special cases to be handled
- Horizontal edges should be excluded
- For vertices lying on scanlines,
- count twice for a change in slope.
- Shorten edge by one scanline for no change in

slope - Coherence between scanlines tells us that
- Edges that intersect scanline y are likely to

intersect y 1 - X changes predictably from scanline y to y 1

- We have 2 data structures Edge Table and Active

Edge Table - Traverse Edges to construct an Edge Table
- Eliminate horizontal edges
- Add edge to linked-list for the scan line

corresponding to the lower vertex. - Store the following
- y_upper last scanline to consider
- x_lower starting x coordinate for edge
- 1/m for incrementing x compute
- Construct Active Edge Table during scan

conversion. AEL is a linked list of active edges

on the current scanline, y. Each active edge line

has the following information - y_upper last scanline to consider
- x_lower edges intersection with current y
- 1/m x increment
- The active edges are kept sorted by x

- Algorithm
- Set y to the smallest y coordinate that has an

entry in the ET i.e, y for the first nonempty

bucket. - Initialize the AET to be empty.
- Repeat until the AET and ET are empty
- 3.1 Move from ET bucket y to the AET those edges

whose y_min y (entering edges). - 3.2 Remove from the AET those entries for which y

y_max (edges not involved in the next

scanline), the sort the AET on x (made easier

because ET is presorted). - 3.3 Fill in desired pixel values on scanline y by

using pairs of x coordinates from AET. - 3.4 Increment y by 1 (to the coordinate of the

next scanline). - 3.5 For each nonvertical edge remaining in the

AET, update x for the new y. - Extensions
- Multiple overlapping polygons priorities
- Color, patterns Z for visibility

(No Transcript)

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